Logarithmic means and double series of Bessel functions

2015 ◽  
Vol 11 (05) ◽  
pp. 1535-1556 ◽  
Author(s):  
Bruce C. Berndt ◽  
Sun Kim
Keyword(s):  
Bessel Functions ◽  
Double Series ◽  
Logarithmic Mean ◽  
Riesz Mean ◽  
Lost Notebook ◽  
Logarithmic Means ◽  
Divisor Problems ◽  
Divisor Sums

In his lost notebook, Ramanujan recorded two identities involving double series of Bessel functions that are closely connected with the classical, unsolved circle and divisor problems. In a series of papers with Zaharescu, the authors proved these identities under various interpretations, as well as Riesz mean analogues. In this paper, logarithmic mean analogues, also involving double series of Bessel functions, are established. Weighted divisor sums involving characters play a central role.

scholarly journals Circle and divisor problems, and double series of Bessel functions

2013 ◽  
Vol 236 ◽  
pp. 24-59 ◽  
Author(s):  
Bruce C. Berndt ◽  
Sun Kim ◽  
Alexandru Zaharescu
Keyword(s):  
Bessel Functions ◽  
Double Series ◽  
Divisor Problems

ON INTEGRAL LOGARITHMIC MEANS OF BLASCHKE PRODUCTS FOR A HALF-PLANE

2018 ◽  
Vol 52 (3 (247)) ◽  
pp. 166-171
Author(s):  
G.V. Mikayelyan ◽  
F.V. Hayrapetyan
Keyword(s):  
Half Plane ◽  
Meromorphic Functions ◽  
Fourier Transforms ◽  
Arbitrary Order ◽  
Blaschke Products ◽  
Logarithmic Mean ◽  
Logarithmic Means

Using the Fourier transforms method for meromorphic functions we characterize the behavior of the integral logarithmic mean of arbitrary order of Blaschke products for the half-plane.

scholarly journals Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Wei-Mao Qian ◽  
Yu-Ming Chu
Keyword(s):  
Unique Solution ◽  
Trigonometric Functions ◽  
Logarithmic Mean ◽  
Double Inequality ◽  
Logarithmic Means ◽  
Inverse Trigonometric Functions

We prove that the double inequalityLp(a,b)<U(a,b)<Lq(a,b)holds for alla,b>0witha≠bif and only ifp≤p0andq≥2and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, wherep0=0.5451⋯is the unique solution of the equation(p+1)1/p=2π/2on the interval(0,∞),U(a,b)=(a-b)/[2arctan⁡((a-b)/2ab)], andLp(a,b)=[(ap+1-bp+1)/((p+1)(a-b))]1/p  (p≠-1,0),L-1(a,b)=(a-b)/(log⁡a-log⁡b)andL0(a,b)=(aa/bb)1/(a-b)/eare the Yang, andpth generalized logarithmic means ofaandb, respectively.

scholarly journals The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Wei-Feng Xia ◽  
Yu-Ming Chu ◽  
Gen-Di Wang
Keyword(s):  
Convex Combination ◽  
Arithmetic Mean ◽  
Power Mean ◽  
Logarithmic Mean ◽  
Positive Real ◽  
Lower Power ◽  
Double Inequality ◽  
Logarithmic Means

Forp∈ℝ, the power meanMp(a,b)of orderp, logarithmic meanL(a,b), and arithmetic meanA(a,b)of two positive real valuesaandbare defined byMp(a,b)=((ap+bp)/2)1/p, forp≠0andMp(a,b)=ab, forp=0,L(a,b)=(b-a)/(log⁡b-log⁡a), fora≠bandL(a,b)=a, fora=bandA(a,b)=(a+b)/2, respectively. In this paper, we answer the question: forα∈(0,1), what are the greatest valuepand the least valueq, such that the double inequalityMp(a,b)≤αA(a,b)+(1-α)L(a,b)≤Mq(a,b)holds for alla,b>0?

scholarly journals A Unitary Treatment of Certain Inequalities Involving Means

2021 ◽  
Vol 45 (02) ◽  
pp. 181-190
Author(s):  
A. F. ALBIŞORU ◽  
M. STROE
Keyword(s):  
Convex Function ◽  
Logarithmic Mean ◽  
Logarithmic Means

The aim of this paper is to state and prove certain inequalities that involve means (e.g., the arithmetic, geometric, logarithmic means) using a particular result. First of all we recall useful properties of a real-valued convex function that will be used in the proof of our inequalities. Further, we present three inequalities, the first involving the logarithmic mean, the second involving the classical arithmetical and geometrical means and in the last we introduce a new mean. Finally, we give alternate proofs to the Schweitzer’s inequality and Khanin’s inequality.

Integral and series representations of q-Polynomials and functions: Part III theta, Ramanujan and q-Bessel functions

2021 ◽  
pp. 1-18
Author(s):  
Mourad E. H. Ismail ◽  
Ruiming Zhang
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Theta Functions ◽  
Bessel Polynomials ◽  
Partition Identity ◽  
Lost Notebook ◽  
Series Representations ◽  
Text Version ◽  
Modified Formula

In this paper, we use an identity connecting a modified [Formula: see text]-Bessel function and a [Formula: see text] function to give [Formula: see text]-versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an [Formula: see text]-version of a partition identity. We prove new relations and identities involving theta functions, the Ramanujan function, the Stieltjes–Wigert, [Formula: see text]-Lommel and [Formula: see text]-Bessel polynomials. We introduce and study [Formula: see text]-analogues of the spherical Bessel functions.

scholarly journals Optimal Lower Generalized Logarithmic Mean Bound for the Seiffert Mean

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Ying-Qing Song ◽  
Wei-Mao Qian ◽  
Yun-Liang Jiang ◽  
Yu-Ming Chu
Keyword(s):  
Logarithmic Mean ◽  
Seiffert Mean ◽  
Logarithmic Means

We present the greatest valuepsuch that the inequalityP(a,b)>Lp(a,b)holds for alla,b>0witha≠b, whereP(a,b)andLp(a,b)denote the Seiffert andpth generalized logarithmic means ofaandb, respectively.

Determination of the Probability Distribution of the Mean Sound Level

2010 ◽  
Vol 35 (4) ◽  
pp. 543-550 ◽  
Author(s):  
Wojciech Batko ◽  
Bartosz Przysucha
Keyword(s):  
Probability Distribution ◽  
Probability Distribution Function ◽  
Mean Value ◽  
Sound Level ◽  
Function Form ◽  
Noise Levels ◽  
Logarithmic Mean ◽  
The Mean ◽  
Estimation Of Uncertainty

AbstractAssessment of several noise indicators are determined by the logarithmic mean <img src="/fulltext-image.asp?format=htmlnonpaginated&src=P42524002G141TV8_html\05_paper.gif" alt=""/>, from the sum of independent random resultsL1;L2; : : : ;Lnof the sound level, being under testing. The estimation of uncertainty of such averaging requires knowledge of probability distribution of the function form of their calculations. The developed solution, leading to the recurrent determination of the probability distribution function for the estimation of the mean value of noise levels and its variance, is shown in this paper.

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